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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.double_exponential.de_levels"></a><a class="link" href="de_levels.html" title="Setting the Maximum Interval Halvings and Memory Requirements">Setting the
      Maximum Interval Halvings and Memory Requirements</a>
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<p>
        The max interval halvings is the maximum number of times the interval can
        be cut in half before giving up. If the accuracy is not met at that time,
        the routine returns its best estimate, along with the <code class="computeroutput"><span class="identifier">error</span></code>
        and <code class="computeroutput"><span class="identifier">L1</span></code>, which allows the
        user to decide if another quadrature routine should be employed.
      </p>
<p>
        An example of this is
      </p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">());</span>
<span class="identifier">size_t</span> <span class="identifier">max_halvings</span> <span class="special">=</span> <span class="number">12</span><span class="special">;</span>
<span class="identifier">tanh_sinh</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span> <span class="identifier">integrator</span><span class="special">(</span><span class="identifier">max_halvings</span><span class="special">);</span>
<span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="number">5</span><span class="special">*</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">7</span><span class="special">;</span> <span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">error</span><span class="special">,</span> <span class="identifier">L1</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">0</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">1</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">error</span><span class="special">,</span> <span class="special">&amp;</span><span class="identifier">L1</span><span class="special">);</span>
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">error</span><span class="special">*</span><span class="identifier">L1</span> <span class="special">&gt;</span> <span class="number">0.01</span><span class="special">)</span>
<span class="special">{</span>
    <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">some_other_quadrature_method</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">0</span><span class="special">,</span> <span class="special">(</span><span class="keyword">double</span><span class="special">)</span> <span class="number">1</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<p>
        It's important to remember that the number of sample points doubles with
        each new level, as does the memory footprint of the integrator object. Further,
        if the integral is smooth, then the precision will be doubling with each
        new level, so that for example, many integrals can achieve 100 decimal digit
        precision after just 7 levels. That said, abscissa-weight pairs for new levels
        are computed only when a new level is actually required (see thread safety),
        none the less, you should avoid setting the maximum arbitrarily high "just
        in case" as the time and space requirements for a large number of levels
        can quickly grow out of control.
      </p>
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